Question

Two identical rods of copper and iron are coated with wax uniformly. When one end of each is kept at temperature of boiling water, the length up to which wax melts are 8.4 cm and 4.2 cm, respectively. If the thermal conductivity of copper is 0.92, then what is the thermal conductivity of iron?
A. 0.23
B. 0.46
C. 0.115
D. 0.69

Answer 1

Hint: Thermal conductivity of any material is defined as its capability to conduct heat. Thermal conductivity depends on material. Mathematically, thermal conductance is defined as the quantity of heat that passes in unit time through a plate of a particular area and thickness when its opposite face has a difference in temperature by one kelvin.

Formula used:
Thermal conductivity, $K=\dfrac{QL}{A\Delta T}$

Complete step by step answer:
Thermal conductivity of a material is its capacity to conduct heat through it. Thermal conductivity depends on the type of material. Heat flows from region of high temperature to region of low temperature. Mathematically, we can write thermal conductivity as
$K=\dfrac{QL}{A\Delta T}$
where $Q$ is the heat transferred through the material, $L$ is the length upto which $\Delta T$ or change in temperature is measured, A is the area of cross-section.
Now, since both rods of iron and copper are identical. This means, Their area of cross-section and length is the same. The temperature difference up to the length the wax melts will be the same.
Thermal conductivity of iron can be written as
${{K}_{iron}}=\dfrac{Q{{L}_{iron}}}{A\Delta T}$
Similarly for copper rod
${{K}_{copper}}=\dfrac{Q{{L}_{copper}}}{A\Delta T}$
Taking ratio of ${{K}_{iron}}$ and ${{K}_{copper}}$, we get
$\dfrac{{{K}_{iron}}}{{{K}_{copper}}}=\dfrac{{{L}_{iron}}}{{{L}_{iron}}}=\dfrac{4.2cm}{8.4cm}=0.5$
$\Rightarrow \dfrac{{{K}_{iron}}}{0.92}=0.5$
$\Rightarrow {{K}_{iron}}=0.46$units of thermal conductivity.
So, the correct answer is “Option B”.

Note:
When a metal bar is heated from one end, with a span of time, a state is reached when the temperature of layers in the metal bar become constant and decrease on moving away from the hot end. This state is known as a steady state. The rate of change of temperature with distance in the direction of flow of thermal current or heat is known as temperature gradient. Thermal conductivity is inversely proportional to temperature gradient.